






def   a_			(   _A	)      ->      int:







				"""simple docstring"""

				if not grid or not grid[0]:
								raise TypeError('The grid does not contain the appropriate information'	)

				for cell_n in range(1   , len(grid[0]	)	):
								grid[0][cell_n] += grid[0][cell_n - 1]
				snake_case__						=  grid[0]

				for row_n in range(1   , len(UpperCAmelCase_	)	):
								snake_case__						=  grid[row_n]
								snake_case__						=  fill_row(UpperCAmelCase_   , UpperCAmelCase_	)
								snake_case__						=  grid[row_n]

				return grid[-1][-1]
def   a_			(   _A   , _A	)      ->      list:







				"""simple docstring"""
				current_row[0] += row_above[0]
				for cell_n in range(1   , len(UpperCAmelCase_	)	):
								current_row[cell_n] += min(current_row[cell_n - 1]   , row_above[cell_n]	)

				return current_row


if __name__ == "__main__":
							import doctest

							doctest.testmod()






"""simple docstring"""


import numpy as np



def 						lowerCamelCase__      (				UpperCAmelCase_		,   UpperCAmelCase_		,   UpperCAmelCase_ = 1E-12		,   UpperCAmelCase_ = 1_00		,   )->				tuple[float, np.ndarray]:







       """simple docstring"""
       assert np.shape(UpperCAmelCase_					)[0] == np.shape(UpperCAmelCase_					)[1]
       # Ensure proper dimensionality.
       assert np.shape(UpperCAmelCase_					)[0] == np.shape(UpperCAmelCase_					)[0]
       # Ensure inputs are either both complex or both real
       assert np.iscomplexobj(UpperCAmelCase_					) == np.iscomplexobj(UpperCAmelCase_					)
       UpperCamelCase       			=		np.iscomplexobj(UpperCAmelCase_					)
       if is_complex:
              # Ensure complex input_matrix is Hermitian
              assert np.array_equal(UpperCAmelCase_		,   input_matrix.conj().T					)

       # Set convergence to False. Will define convergence when we exceed max_iterations
       # or when we have small changes from one iteration to next.

       UpperCamelCase       			=		False
       UpperCamelCase       			=		0
       UpperCamelCase       			=		0
       UpperCamelCase       			=		1E12

       while not convergence:
              # Multiple matrix by the vector.
              UpperCamelCase       			=		np.dot(UpperCAmelCase_		,   UpperCAmelCase_					)
              # Normalize the resulting output vector.
              UpperCamelCase       			=		w / np.linalg.norm(UpperCAmelCase_					)
              # Find rayleigh quotient
              # (faster than usual b/c we know vector is normalized already)
              UpperCamelCase       			=		vector.conj().T if is_complex else vector.T
              UpperCamelCase       			=		np.dot(UpperCAmelCase_		,   np.dot(UpperCAmelCase_		,   UpperCAmelCase_					)					)

              # Check convergence.
              UpperCamelCase       			=		np.abs(lambda_ - lambda_previous					) / lambda_
              iterations += 1

              if error <= error_tol or iterations >= max_iterations:
                     UpperCamelCase       			=		True

              UpperCamelCase       			=		lambda_

       if is_complex:
              UpperCamelCase       			=		np.real(lambda_					)

       return lambda_, vector



def 						lowerCamelCase__      (				)->				None:







       """simple docstring"""
       UpperCamelCase       			=		np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]					)
       UpperCamelCase       			=		np.array([41, 4, 20]					)
       UpperCamelCase       			=		real_input_matrix.astype(np.complexaaa					)
       UpperCamelCase       			=		np.triu(1j * complex_input_matrix		,   1					)
       complex_input_matrix += imag_matrix
       complex_input_matrix += -1 * imag_matrix.T
       UpperCamelCase       			=		np.array([41, 4, 20]					).astype(np.complexaaa					)

       for problem_type in ["real", "complex"]:
              if problem_type == "real":
                     UpperCamelCase       			=		real_input_matrix
                     UpperCamelCase       			=		real_vector
              elif problem_type == "complex":
                     UpperCamelCase       			=		complex_input_matrix
                     UpperCamelCase       			=		complex_vector

              # Our implementation.
              UpperCamelCase			,							UpperCamelCase       			=		power_iteration(UpperCAmelCase_		,   UpperCAmelCase_					)

              # Numpy implementation.

              # Get eigenvalues and eigenvectors using built-in numpy
              # eigh (eigh used for symmetric or hermetian matrices).
              UpperCamelCase			,							UpperCamelCase       			=		np.linalg.eigh(UpperCAmelCase_					)
              # Last eigenvalue is the maximum one.
              UpperCamelCase       			=		eigen_values[-1]
              # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
              UpperCamelCase       			=		eigen_vectors[:, -1]

              # Check our implementation and numpy gives close answers.
              assert np.abs(eigen_value - eigen_value_max					) <= 1E-6
              # Take absolute values element wise of each eigenvector.
              # as they are only unique to a minus sign.
              assert np.linalg.norm(np.abs(UpperCAmelCase_					) - np.abs(UpperCAmelCase_					)					) <= 1E-6


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      test_power_iteration()





